Cardano and the Exclusive Club

In a small village there is an Exclusive Club, consisting of every
person who doesn't shave anyone who shaves him. There is also
the local barber, Cardano, who boast that he shaves the Exclusive
Club, the whole Exclusive Club, and nothing but the Exclusive Club.
Prove that Cardano cannot be telling the truth.

Representation

We write x and y for arbitrary people in the village, and let c stand for Cardano.
We use the following notations:

xSy == "x shaves y"
E.x == "x is a member of the Exclusive Club"

The definition of the Exclusive Club can be represented as

(1) (Ax|: E.x == (Ay| ySx: !xSy))

and Cardano's claim is

(2) (Ax|: cSx == E.x)

Solution

We will prove that (1) and (2) together lead to a contradiction. We start
with (1): for every x,

So from (1) and (2) we've derived both that Cardano shaves himself, and
that he doesn't, which is a contradiction. Therefore Cardano was lying.
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